Non-Zero-Sum Games

Jennifer Firkins Nordstrom · Introduction to Game Theory, Section 4.1 · CC BY-SA 4.0

When payoffs don't sum to zero, both players can win or both can lose. Cooperation enters. The zero-sum tools (minimax, graphical method, expected value method) break down because minimizing the opponent's payoff no longer maximizes yours.

Zero-sum vs non-zero-sum

In a zero-sum game, every dollar you gain is a dollar your opponent loses. In a non-zero-sum game, the total payoff can grow or shrink. Trade is positive-sum (both sides gain). War is negative-sum (both sides lose). Most real situations are non-zero-sum.

Scheme

; Zero-sum: payoffs always sum to 0
(define (zero-sum? matrix)
  (every (lambda (row)
    (every (lambda (cell)
      (= 0 (+ (car cell) (cadr cell))))
    row))
  matrix))

; Zero-sum game: [[3,-3],[-2,2]],[[-1,1],[4,-4]]
(define zs-game '(((3 -3) (-2 2)) ((-1 1) (4 -4))))
(display "Zero-sum? ") (display (zero-sum? zs-game)) (newline)

; Battle of the Sexes: (2,1),(-1,-1),(-1,-1),(1,2)
(define bos-game '(((2 1) (-1 -1)) ((-1 -1) (1 2))))
(display "Zero-sum? ") (display (zero-sum? bos-game)) (newline)

; Check totals
(display "BoS cell sums: ")
(for-each (lambda (row)
  (for-each (lambda (cell)
    (display (+ (car cell) (cadr cell))) (display " "))
  row))
bos-game)
; 3, -2, -2, 3 -- not constant

Communication changes everything

In a zero-sum game, telling your opponent your strategy is always bad. In a non-zero-sum game, communication can help both players. Battle of the Sexes: if Alice and Bob can talk, they coordinate on one movie and both benefit. Without communication, they might clash and both get -1.

Scheme

; Battle of the Sexes: coordination matters
;           Action  Comedy
;  Action  (2, 1)  (-1,-1)
;  Comedy (-1,-1)  (1, 2)
;
; Two Nash equilibria: (Action, Action) and (Comedy, Comedy)
; Without communication: might end up at (-1, -1)
; With communication: agree on one, both benefit

(define (bos-payoff p1 p2)
  (cond ((and (eq? p1 'A) (eq? p2 'A)) '(2 1))
        ((and (eq? p1 'C) (eq? p2 'C)) '(1 2))
        (else '(-1 -1))))

(display "Both pick Action:  ") (display (bos-payoff 'A 'A)) (newline)
(display "Both pick Comedy:  ") (display (bos-payoff 'C 'C)) (newline)
(display "Mismatch:          ") (display (bos-payoff 'A 'C)) (newline)

; In zero-sum, revealing "I'll play A" lets opponent exploit you.
; Here, revealing "I'll go to Action" helps both coordinate.
(display "Communication helps: both outcomes beat the mismatch")

Why zero-sum tools break

The graphical method and expected value method both assume that minimizing the opponent's payoff is the same as maximizing yours. In non-zero-sum games, this is false. You might gain by helping your opponent gain. New tools needed: best response analysis, dominance with payoff pairs, and eventually the full Nash equilibrium machinery.

Scheme

; In zero-sum: maximin = minimax (von Neumann)
; Your best strategy against their worst case IS the equilibrium.
;
; In non-zero-sum: maximin is NOT the answer.
; Battle of the Sexes maximin analysis:

(define (p1-maximin)
  ; Row A: payoffs 2, -1. Minimum = -1.
  ; Row C: payoffs -1, 1. Minimum = -1.
  ; Maximin: both rows tie at -1. No help.
  (display "P1 maximin: min(A) = -1, min(C) = -1") (newline)
  (display "  Maximin says: indifferent. Not useful.") (newline))

(p1-maximin)

; But Nash equilibrium finds two pure equilibria:
(display "Nash equilibria: (A,A) and (C,C)") (newline)
(display "  Both are stable: no unilateral deviation helps.") (newline)
(display "  Maximin missed them because it ignores coordination.")

Notation reference

Concept	Zero-sum	Non-zero-sum
Payoff cell	(a, -a)	(a, b) where a + b varies
Opponent's loss	= your gain	independent
Communication	always hurts	can help (coordination)
Maximin	= equilibrium	too conservative
Equilibria	unique (mixed)	often multiple

Neighbors

Nordstrom sequence

🎲 Nordstrom 3.3 — Mixed Strategies: Expected Value (previous)
🎲 Nordstrom 4.2 — Prisoner's Dilemma and Chicken (next)

🎲 Nordstrom 2.1 — Zero-Sum Games (the contrast)
🍞 Hedges 2018 — compositional game theory handles non-zero-sum naturally

Foundations (Wikipedia)

Translation notes

The Battle of the Sexes example is from Nordstrom's textbook. The shift from zero-sum to non-zero-sum is not just a change in payoff numbers. It's a change in the strategic structure: communication, cooperation, and multiple equilibria all become relevant. This is why Chapters 2-3 (zero-sum) needed different tools from Chapter 4 onward.

Want the full treatment? Read Section 4.1 in Nordstrom's textbook.

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